Table of Contents
ISRN Algebra
Volume 2011 (2011), Article ID 104263, 9 pages
http://dx.doi.org/10.5402/2011/104263
Research Article

Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups

Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut 11072809, Lebanon

Received 17 August 2011; Accepted 5 September 2011

Academic Editors: A. V. Kelarev, A. Kiliçman, and A. Rapinchuk

Copyright © 2011 Mohammad N. Abdulrahim and Wiaam M. Zeid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The reduced Burau representation is a one-parameter representation of 𝐡 𝑛 , the braid group on 𝑛 strings. Specializing the parameter to nonzero complex number π‘₯ gives a representation 𝛽 𝑛 ( π‘₯ ) : 𝐡 𝑛 β†’ 𝐺 𝐿 ( β„‚ 𝑛 βˆ’ 1 ) , which is either irreducible or has an irreducible composition factor Μ‚ β€Œ 𝛽 𝑛 ( π‘₯ ) : 𝐡 𝑛 β†’ 𝐺 𝐿 ( β„‚ 𝑛 βˆ’ 2 ) . In our paper, we let π‘˜ β‰₯ 2 , and we determine a sufficient condition for the irreducibility of the tensor product of π‘˜ irreducible Burau representations. This is a generalization of our previous work concerning the cases π‘˜ = 2 and π‘˜ = 3 .